# Find the Concavity F(x)=x square root of 9-x Find the inflection points.
Find the second derivative.
Find the first derivative.
Rewrite as .
Differentiate using the Product Rule which states that is where and .
Differentiate using the chain rule, which states that is where and .
To apply the Chain Rule, set as .
Differentiate using the Power Rule which states that is where .
Replace all occurrences of with .
To write as a fraction with a common denominator, multiply by .
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Combine.
Multiply by .
Combine the numerators over the common denominator.
Simplify the numerator.
Multiply by .
Subtract from .
Combine fractions.
Move the negative in front of the fraction.
Combine and .
Move to the denominator using the negative exponent rule .
Combine and .
By the Sum Rule, the derivative of with respect to is .
Since is constant with respect to , the derivative of with respect to is .
Since is constant with respect to , the derivative of with respect to is .
Combine fractions.
Combine and .
Simplify the expression.
Move to the left of .
Rewrite as .
Move the negative in front of the fraction.
Differentiate using the Power Rule which states that is where .
Multiply by .
Differentiate using the Power Rule which states that is where .
Multiply by .
To write as a fraction with a common denominator, multiply by .
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Combine.
Multiply by .
Combine the numerators over the common denominator.
Multiply by by adding the exponents.
Move .
Use the power rule to combine exponents.
Combine the numerators over the common denominator.
Divide by .
Simplify .
Move to the left of .
Simplify.
Apply the distributive property.
Simplify the numerator.
Simplify each term.
Multiply by .
Multiply by .
Subtract from .
Factor out of .
Rewrite as .
Factor out of .
Rewrite as .
Move the negative in front of the fraction.
Find the second derivative.
Since is constant with respect to , the derivative of with respect to is .
Differentiate using the Quotient Rule which states that is where and .
Multiply the exponents in .
Apply the power rule and multiply exponents, .
Cancel the common factor of .
Cancel the common factor.
Rewrite the expression.
Simplify.
Differentiate.
By the Sum Rule, the derivative of with respect to is .
Since is constant with respect to , the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
Multiply by .
Since is constant with respect to , the derivative of with respect to is .
Simplify the expression.
Move to the left of .
Differentiate using the chain rule, which states that is where and .
To apply the Chain Rule, set as .
Differentiate using the Power Rule which states that is where .
Replace all occurrences of with .
To write as a fraction with a common denominator, multiply by .
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Combine.
Multiply by .
Combine the numerators over the common denominator.
Simplify the numerator.
Multiply by .
Subtract from .
Combine fractions.
Move the negative in front of the fraction.
Combine and .
Move to the denominator using the negative exponent rule .
By the Sum Rule, the derivative of with respect to is .
Since is constant with respect to , the derivative of with respect to is .
Since is constant with respect to , the derivative of with respect to is .
Multiply.
Multiply by .
Multiply by .
Differentiate using the Power Rule which states that is where .
Combine fractions.
Multiply by .
Multiply and .
Move to the left of .
Simplify.
Apply the distributive property.
Simplify the numerator.
Simplify each term.
Multiply and .
Factor out of .
Factor out of .
Factor out of .
Factor out of .
To write as a fraction with a common denominator, multiply by .
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Combine.
Multiply by .
Combine the numerators over the common denominator.
Simplify the numerator.
Factor out of .
Factor out of .
Factor out of .
Rewrite using the commutative property of multiplication.
Multiply by by adding the exponents.
Move .
Use the power rule to combine exponents.
Combine the numerators over the common denominator.
Divide by .
Simplify .
Apply the distributive property.
Multiply by .
Multiply by .
Subtract from .
Factor out of .
Rewrite as .
Factor out of .
Rewrite as .
Move the negative in front of the fraction.
Combine terms.
Multiply by .
Multiply by .
Rewrite as a product.
Multiply and .
Move to the left of .
Multiply by .
Multiply by .
Reorder terms.
The second derivative of with respect to is .
Set the second derivative equal to then solve the equation .
Simplify the denominator.
Factor out of .
Factor out of .
Factor out of .
Factor out of .
Combine exponents.
Multiply by .
Raise to the power of .
Use the power rule to combine exponents.
To write as a fraction with a common denominator, multiply by .
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Combine.
Multiply by .
Combine the numerators over the common denominator.
Simplify the numerator.
Multiply by .
Find the LCD of the terms in the equation.
Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.
The LCM is the smallest number that all of the numbers divide into evenly.
1. List the prime factors of each number.
2. Multiply each factor the greatest number of times it occurs in either number.
has factors of and .
The number is not a prime number because it only has one positive factor, which is itself.
Not prime
The LCM of is the result of multiplying all prime factors the greatest number of times they occur in either number.
Multiply by .
The factors for are , which is multiplied by itself times.
occurs times.
The LCM of is the result of multiplying all factors the greatest number of times they occur in either term.
The Least Common Multiple of some numbers is the smallest number that the numbers are factors of.

Simplify the left side of the equation by cancelling the common factors.
Rewrite using the commutative property of multiplication.
Cancel the common factor of .
Cancel the common factor.
Rewrite the expression.
Cancel the common factor of .
Cancel the common factor.
Rewrite the expression.
Apply the distributive property.
Multiply by .
Multiply .
Multiply by .
Multiply by .
Solve the equation.
Add to both sides of the equation.
Divide each term by and simplify.
Divide each term in by .
Cancel the common factor of .
Cancel the common factor.
Divide by .
Divide by .
Exclude the solutions that do not make true.
No solution
No solution
No values found that can make the second derivative equal to .
No Inflection Points
No Inflection Points
Find the domain of .
Set the radicand in greater than or equal to to find where the expression is defined.
Solve for .
Subtract from both sides of the inequality.
Multiply each term in by
Multiply each term in by . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.
Multiply .
Multiply by .
Multiply by .
Multiply by .
The domain is all values of that make the expression defined.
Interval Notation:
Set-Builder Notation:
Interval Notation:
Set-Builder Notation:
Create intervals around the inflection points and the undefined values.
Substitute any number from the interval into the second derivative and evaluate to determine the concavity.
Replace the variable with in the expression.
Simplify the result.
Rewrite as .
Factor out of .
Factor out of .
Reorder terms.
Cancel the common factors.
Factor out of .
Cancel the common factor.
Rewrite the expression.
Factor out of .
Factor out of .
Factor out of .
Cancel the common factors.
Factor out of .
Cancel the common factor.
Rewrite the expression.
Factor out of .
Factor out of .
Factor out of .
Cancel the common factors.
Factor out of .
Cancel the common factor.
Rewrite the expression.
Subtract from .
Simplify the denominator.
Combine exponents.
Factor out negative.
Multiply by .
Multiply by .
Rewrite as .
Apply the power rule and multiply exponents, .
Cancel the common factor of .
Cancel the common factor.
Rewrite the expression.
Evaluate the exponent.
Simplify the denominator.
Multiply by .
Multiply by .
Move the negative in front of the fraction.
The final answer is .
The graph is concave down on the interval because is negative.
Concave down on since is negative
Concave down on since is negative
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